RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 19 



Here the function /which determines the stress in the strata parallel to xy is 



Now, this function is not sufficiently general, for instead of being any function of 

 x, y and z, it is the product of a function of x and y multiplied by a function of z. 

 Besides this, though the value of p zz is, as it ought to be, a function of x and 

 y only, it is not of the most general form, for it depends on G, the function which 

 determines the stresses^, p xy , andp^, whereas the value oip zz may be entirely 

 independent of the values of these stresses. In fact, the equations give 



„ „ PxxPvy — Pxy 



Pzz ~ P fd 2 Z\* 



This method, therefore, of representing stress in a body of three dimensions 

 is a restricted solution of the equations of equilibrium. 



On Reciprocal Diagrams in Three Dimensions. 



Let us consider figures in two portions of space, which we shall call respec- 

 tively the first and the second diagrams. Let the co-ordinates of any point in 

 the first diagram be denoted by x, y, z, and those of the corresponding point in 

 the second by £, 17, £, measured in directions parallel to x, y, z respectively. 

 Let F be a quantity varying from point to point of the first figure in any con- 

 tinuous manner ; that is to say, if A, B are two points, and F 1 , F 2 the values 

 of F at those points ; then, if B approaches A without limit, the value of F 2 

 approaches that of F x without limit. Let the co-ordinates (£, 77, £) of a point in 

 the second diagram be determined from x, y, z, those of the corresponding point 

 in the first by the equations 



£F _ dF dF 



? ~~ dx ' dy ' ' ~ dz ' ^ '' 



This is equivalent to the statement, that the vector (p) of any point in the 

 second diagram represents in direction and magnitude the rate of variation of F 

 at the corresponding point of the first diagram. 



Next, let us determine another function, <f>, from the equation 



xZ + i JV + zZ=F+ <j> (2), 



<£, as thus determined, will be a function of x, y, and z, since £, tj, £ are known 

 in terms of these quantities. But, for the same reason, </> is a function of £ 17, £. 

 Differentiate <f> with respect to £, considering x, y and z functions of £ 77, £, 



deb „ dx dy . dz dF 



-Z-x + Z-^+V-JL + t;-^-- 



